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The Catalan numbers are a sequence of natural numbers that occur in various counting problems, often involving recursively defined objects. They are named after Eugène Catalan, though they were previously discovered in the 1730s by Minggatu. The n-th Catalan number can be expressed directly in terms of the central binomial coefficients by
This number is given by the 5th Catalan number. It is trivial to triangulate any convex polygon in linear time into a fan triangulation, by adding diagonals from one vertex to all other non-nearest neighbor vertices. The total number of ways to triangulate a convex n-gon by non-intersecting diagonals is the (n−2)nd Catalan number, which equals
The number of noncrossing partitions of a set of n elements is the nth Catalan number. The number of noncrossing partitions of an n -element set with k blocks is found in the Narayana number triangle.
Catalan's trapezoids are a countable set of number trapezoids which generalize Catalan’s triangle. Catalan's trapezoid of order m = 1, 2, 3, ... is a number trapezoid whose entries (,) give the number of strings consisting of n X-s and k Y-s such that in every initial segment of the string the number of Y-s does not exceed the number of X-s by m or more. [6]
Substituting k = 1 into this formula gives the Catalan numbers and substituting k = 2 into this formula gives the Schröder–Hipparchus numbers. [7] In connection with the property of Schröder–Hipparchus numbers of counting faces of an associahedron, the number of vertices of the associahedron is given by the Catalan numbers.
The Fuss-Catalan represents the number of legal permutations or allowed ways of arranging a number of articles, that is restricted in some way. This means that they are related to the Binomial Coefficient. The key difference between Fuss-Catalan and the Binomial Coefficient is that there are no "illegal" arrangement permutations within Binomial ...
In combinatorial mathematics, the Lobb number L m,n counts the ways that n + m open parentheses and n − m close parentheses can be arranged to form the start of a valid sequence of balanced parentheses. [1] Lobb numbers form a natural generalization of the Catalan numbers, which count the complete strings of balanced parentheses of a given ...
The number of different binary trees on nodes is , the th Catalan number (assuming we view trees with identical structure as identical). For large n {\displaystyle n} , this is about 4 n {\displaystyle 4^{n}} ; thus we need at least about log 2 4 n = 2 n {\displaystyle \log _{2}4^{n}=2n} bits to encode it.